Volume 29, Issue 1. On the empirical relevance of st. petersburg lotteries. James C. Cox Georgia State University

Size: px

Start display at page:

Download "Volume 29, Issue 1. On the empirical relevance of st. petersburg lotteries. James C. Cox Georgia State University"

Jeffrey Warren
5 years ago
Views:

1 Volume 29, Issue 1 O the empirical relevace of st. petersburg lotteries James C. Cox Georgia State Uiversity Vjollca Sadiraj Georgia State Uiversity Bodo Vogt Uiversity of Magdeburg Abstract Expected value theory has bee kow for ceturies to be subject to critique by St. Petersburg paradox argumets. Ad there is a traditioal rebuttal of the critique that deies the empirical relevace of the paradox because of its apparet depedece o existece of credible offers to pay ubouded sums of moey. Neither critique or rebuttal focus o the questio with empirical relevace: Do people make choices i bouded St. Petersburg games that are cosistet with expected value theory? This paper reports a experimet that addresses that questio. The Natioal Sciece Foudatio provided research support (grat umber IIS ). Citatio: James C. Cox ad Vjollca Sadiraj ad Bodo Vogt, (2009) ''O the empirical relevace of st. petersburg lotteries'', Ecoomics Bulleti, Vol. 29 o.1 pp Submitted: Ja Published: February 27, 2009.

2 1. Itroductio The first theory of decisio uder risk, expected value maximizatio, was challeged log ago by the St. Petersburg Paradox (Beroulli 1738). 1 The origial St. Petersburg lottery pays 2 whe a fair coi comes up heads for the first time o flip, a evet with probability 1/ 2. The expected value of this lottery is ifiite (that is, larger tha ay fiite umber) because = 1 2 (1/2) = But Beroulli famously reported that most people stated they would be uwillig to pay more tha a small amout to play this lottery. He cocluded that such reported prefereces called ito questio the validity of expected value theory, ad offered a theory with decreasig margial utility of moey as a replacemet. A traditioal rebuttal of the alleged paradox is based o the observatio that o aget could credibly offer the St. Petersburg lottery for aother to play because it could result i a payout obligatio exceedig ay aget s wealth ad therefore that this challege to expected value theory has o bite. Suppose, for example, that the maximum prize offered is 35 equal to $2 (= $34.36 billio), a amout that Exxo Mobil could have credibly offered to pay i 2008 sice its reported profit for 2007 was $40 billio. Cosider a St. Petersburg lottery that pays $2 35 if the first head occurs o flip, for 35, ad pays $2 i the evet that there is a ru of 35 tails i the first 35 coi flips. This lottery has expected value of $36 35 = [$2 (1/2) ] + $2 (1/2) = , so it would ot be paradoxical if idividuals stated they would be uwillig to pay large amouts to play the lottery. A alterative versio of the lottery pays $2 if the first head occurs o flip, for 35, ad othig if there is a ru of 35 tails i the first 35 coi flips. The expected value of this lottery is $35; a expected value maximizig aget would ot pay more tha $35 to play it. Argumets about the St. Petersburg Paradox are great sport, especially ow that a geeralized form of the paradox has bee show (by Cox ad Sadiraj 2008, ad Rieger ad Wag 2006) to apply to cumulative prospect theory (Tversky ad Kahema 1992), dual theory of expected utility theory (Yaari 1987), rak depedet utility theory (Quiggi 1993), ad expected utility theory as well as expected value theory. But either side of such argumets provides a aswer to the questio that is relevat to positive ecoomics: Do real offers to play St. Petersburg lotteries elicit real resposes that are icosistet with expected value theory? Surprisigly, there appears to have bee o previous experimet that addresses this questio A Real Experimet with Fiite St. Petersburg Lotteries The experimet was desiged as follows. Subjects were offered the opportuity to decide whether to pay their ow moey to play ie fiite St. Petersburg lotteries. Oe of each subject s decisios was radomly selected for real moey payoff. Lottery N had a maximum of N coi tosses ad paid 2 euros if the first head occurred o toss umber, 1 Aalysis of the paradox was published by Daiel Beroulli (1738). It was, reportedly, previously described i a letter dated September 9, 1713 from Nicolas Beroulli to Pierre Raymod de Motmort. 2 Beroulli (1738), Bottom, et al. (1989), ad Rivero, et al. (1990) report experimets with hypothetical resposes. After our experimet was completed (i February 2007) we became aware of research o St. Petersburg lotteries by Tibor Neugebauer from his presetatio at the North America Regioal Coferece of the Ecoomic Sciece Associatio, November 13-15, As of the date of this writig, Professor Neugebauer s workig paper is ot yet available. 1

3 for = 1,2,... N, ad paid othig if o head occurred. Lotteries were offered for N = 1,2,,9. Of course, the expected payoff from playig lottery N was N euros. The price offered to a subject for playig lottery N was 25 euro cets lower tha N euros. Expected value theory predicts preferece for the St. Petersburg lottery N for every oe of these lotteries, i.e. for all N = 1,,9. The experimet was ru i the Maxlab at the Uiversity of Magdeburg i February At the begiig of the experimet, subjects were seated at well-separated cubicles ad were give prited subject istructios. After readig the istructios, each subject made ie decisios. Subsequetly, the decisio sheets were collected. Next, oe ball was draw, for each idividual subject, from a bigo cage cotaiig ie balls. The umber o the ball draw for a subject selected the decisio that paid moey for that subject. If a subject had chose the lottery i the radomly selected decisio, moey payoff i the lottery was determied by repeatedly flippig a coi util either the first head appeared or util the specified maximum umber of tosses i the selected lottery had bee attaied. Bigo balls were draw ad the coi was flipped i the presece of the subjects. Each subject was paid the amout determied by this procedure i cash, i Euros, immediately at the ed of the experimet. A Eglish versio of the subject istructios is icluded i the appedix. 3. Do People Make Risk Neutral Choices with Fiite St. Petersburg Lotteries? Thirty subjects participated i the experimet: (a) 26 out of 30 (or 87%) of the idividual subjects refused at least oe opportuity to play a St. Petersburg lottery for less tha its expected value; ad (b) over all subjects, 127 out of 270 (or 47%) of their choices were icosistet with expected value theory. We ext ask whether the observed failure of expected value theory is statistically sigificat. Figure 1. Proportios of Subjects Who Rejected St. Petersburg Lotteries 2

4 Oe way to pose the questio is to ask which characterizatio of risk refereces is more cosistet with the data: (a) risk eutrality; or (b) risk aversio sufficiet to imply rejectio of all offers to play the St. Petersburg games i the experimet. First, the observed proportios of subjects who reject St. Petersburg lotteries for N = 1,2, or 9 are: [0.13, 0.13, 0.20, 0. 37, 0.47, 0.57, 0.73, 0.80, 0.83], as show i Figure 1. I the first 5 tasks more tha half the subjects made choices that are cosistet with expected value (EV) theory. However, as the stakes of the sure amout of moey required for playig the St. Petersburg lotteries icrease, ad the variace of payoffs of the lotteries icrease, subjects risk-averse choices start to domiate. From lotteries for N = 6 to 9, the fractio of choices violatig EV theory icreases from 57% to 83%. Next, we apply a liear mixture model (Harless ad Camerer 1994) with stochastic preferece specificatio for error rate ε : (a) if optio Z is stochastically preferred the Prob(choose Z ) = 1 ε ; ad (b) if optio Z is ot preferred the Prob(choose Z ) = ε. Let the letter a deote a subject s respose that she accepts the offer to play a specific St. Petersburg lottery i the experimet. Let r deote rejectio of the offer to play the game. The liear mixture model is used to address the specific questio whether, for the ie St. Petersburg lotteries offered to the subjects, the respose patter ( aaaaaaaaa,,,,,,,, ) or the respose patter ( rrrrrrrrr,,,,,,,, ) is more cosistet with the data. With this specificatio, the log-likelihood is -182 ad the estimate of the error rate is The poit estimate of the proportio of subjects i the experimet that are ot risk eutral (or very slightly risk averse) is 0.48, ad the Wald 90% cofidece iterval of this estimate is (0.30, 0.67). Allowig for a specificatio with two error rates (oe error rate for lotteries 1-4 ad aother error rate for lotteries 5-9), the estimates of the error rates are 0.50 for the first four lotteries ad 0.14 for the last five lotteries. The poit estimate of the proportio of subjects i the experimet that are ot risk eutral (or very slightly risk averse) is 0.77, with Wald 90% cofidece iterval of (0.63, 0.91). The log-likelihood is Usig data oly for lotteries 4-9, that require paymets i excess of 3 euros to accept, the error rate is 0.18 ad the poit estimate of the proportio of subjects i the experimet that are ot risk eutral (or very slightly risk averse) is 0.71 with 90% cofidece iterval (0.56, 0.87). The log-likelihood is Table I summarizes these results. Table I. Estimates for Error-rate Aalysis Model 1 Model 2 Model 3 (lotteries 4-9) obs Estimated fractio of risk averse subjects 0.48 (0.30, 0.67) 0.77 (0.63,0.91) 0.71 (0.56,0.87) Estimated error rate Estimated error rate Log-likelihood Figures i paretheses show Wald 90% cofidece itervals. 4. Cocludig Remarks The St. Petersburg Paradox has bee kow for almost 300 years. Argumets about its relevace to judgig the plausibility of expected value theory are almost as old. Although argumets about the paradox are great sport, they do t address the origial questio that motivated its itroductio: Are decisio makers risk eutral? We report what appears to be 3

5 the first experimet with real moey payoffs for fiite St. Petersburg lotteries. Data from our experimet support the coclusio that a majority of subjects i the experimet are risk averse, ot risk eutral. Refereces Beroulli, D. (1738) Specime Theoriae Novae de Mesura Sortis Commetarii Academiae Scietiarum Imperialis Petropolitaae, 5, Eglish traslatio: Expositio of a New Theory o the Measuremet of Risk Ecoometrica 22, 1954, Bottom, W.P., R.N. Botempo ad D.R. Holtgrave (1989) Experts, Novices, ad the St. Petersburg Paradox: Is Oe Solutio Eough? Joural of Behavioral Decisio Makig 2, Cox, J.C. ad V. Sadiraj (2008) Risky Decisios i the Large ad i the Small: Theory ad Experimet i Risk Aversio i Experimets by J.C. Cox ad G.W. Harriso, Eds., Emerald: Bigley, UK, Research i Experimetal Ecoomics, Volume 12, Harless, D. ad C. F. Camerer (1994) The Predictive Utility of Geeralized Expected Utility Theories Ecoometrica 62, Quiggi, J. (1993) Geeralized Expected Utility Theory. The Rak-Depedet Model, Kluwer: Bosto. Rieger, M.O. ad M. Wag (2006) Cumulative Prospect Theory ad the St. Petersburg Paradox Ecoomic Theory 28, Rivero, J.C., D.R. Holtgrave, R.N. Botempo ad W.P. Bottom (1990) The St. Petersburg Paradox: Data, At Last Commetary 8, Tversky, A. ad D. Kahema (1992) Advaces i Prospect Theory: Cumulative Represetatio of Ucertaity Joural of Risk ad Ucertaity 5, Yaari, M.E. (1987) The Dual Theory of Choice uder Risk Ecoometrica 55,

6 Appedix: Subject Istructios Please write your idetificatio code here: A coi is tossed o more tha 9 times. Your payoff depeds o the umber of tosses util Head appears for the first time. If Head appears for the first time o flip umber N the you are paid 2 N Euros. Table A.I shows the possible outcomes: Head appears for the first time o coi toss Table A.I Probability this will occur Your payoff i Euros Never 0 There are a variety of differet lotteries offered to you that differ i the maximum possible umber of coi tosses ad the amout you have to pay if you wat to participate i the lottery. Table A.II shows this. Table A.II Maximum umber of Participatio fee i Euros tosses I choose to pay to participate: Yes/No For example, if you decide to pay 3.75 Euros to participate i the lottery with a maximum of 4 tosses, the coi will be flipped 4 times. Your payoff is determied accordig to Table A.I. 5

7 If Head appears o the first toss the you will receive 2 Euros, regardless of the results of the further tosses. If Tail appears o the first toss ad Head o the secod, you will receive 4 Euros, regardless the results of the further tosses. If Tails appear o the first two tosses ad the Head o the third toss, you will receive 8 Euros, regardless of the further tosses. If Tails appear o the first three tosses, followed by Head o the fourth toss, you receive 16 Euros. If Head ever appears your payoff is 0 Euro. Payoffs After you make your decisios, oe of the rows will be selected by chace ad your Yes or No decisio i that row will become bidig. The selectio of the row is carried out by drawig a ball from a bigo cage cotaiig balls with umbers 1,2,, 9. The umber o the draw ball determies the row of the table that is selected. If, for example, row 6 is selected for payoff the: (a) if your decisio i row 6 is No the o moey chages hads; (b) if your decisio i row 6 is Yes the you will pay the experimeter 5.75 Euros to play the coi toss lottery with a maximum of 6 tosses ad possible outcomes i the first 6 rows of Table A.I. 6

On the Empirical Relevance of St.Petersburg Lotteries By James C. Cox, Vjollca Sadiraj, and Bodo Vogt*

On the Empirical Relevance of St.Petersburg Lotteries By James C. Cox, Vjollca Sadiraj, and Bodo Vogt* O the Empirical Relevace of St.Petersburg Lotteries By James C. Cox, Vjollca Sadiraj, ad Bodo Vogt* Expected value theory has bee kow for ceturies to be subject to critique by St. Petersburg paradox argumets.